Hook modifier falloff not working
Falloff of the influence of a Hook does work but it requires some unexpected setup; you need to Assign multiple verticies to the modifier in Edit mode rather than rely on the 'Vertex Group' to indicate the affectedvertices.
Hooking to a single vert will move just that vertex. However, in Edit mode you can also 'Assign' other vertices to the modifier - it is these 'assigned' vertices that are affected by the 'Falloff' of the modifier. A Radius of zero inicates 'no falloff' (so all vertices are affected equally). Therefore, the steps you require are to select and 'assign' the surrounding vertices and then increase the Radius and Type until the required effect is achieved. This is demonstrated in the following image :